A characterization of compacta which admit acyclic $UV^{n-1}$ -resolutions

Abstract

Let $R$ be a commutative ring with identity. The main result of the paper is the following: THEOREM. Let $f:Z\rightarrow X$ be a $UV^{n-1}$-mapping from a compactum $Z$ of dimension $\leq n$ onto a compactum $X$. If $H^{n}(f^{-1}(x);R)=0$ for all $x\in X$, then $a-dim_{R}X\leq n$. As its consequence, we have a characterization of compacta $X$ of $a-dim_{R}X\leq n$. THEOREM. A compactum $X$ admits a $UV^{n-1}$ -mapping $f:Z\rightarrow X$ from a compactum $Z$ of dimension $\leq n$ onto $X$ such that $H^{n}(f^{-1}(x);R)=0$ for all $x\in X$ if and only if $a-dim_{R}X\leq n$.